Advertisement

Ukrainian Mathematical Journal

, Volume 62, Issue 8, pp 1333–1338 | Cite as

On an estimate for the rearrangement of a function from the Muckenhoupt class A 1

  • E. Yu. Leonchik
Article

We obtain an exact estimate for a nonincreasing uniform rearrangement of a function of two variables from the Muckenhoupt class A 1.

Keywords

Singular Integral Exact Estimate Multidimensional Case Bound Analytic Differential Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    B. Muckenhoupt, “Weighted norm inequalities for the Hardy maximal function,” Trans. Amer. Math. Soc., 165, 207–226 (1972).zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    R. Hunt, B. Muckenhoupt, and R. L. Wheeden, “Weighted norm inequalities for the conjugate function and Hilbert transform,” Trans. Amer. Math. Soc., 176, 227–251 (1973).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    A. Korenovskii, “Mean oscillations and equimeasurable rearrangements of functions,” Lect. Notes Unione Mat. Ital., No. 4 (2007).Google Scholar
  4. 4.
    E. Yu. Leonchik, “On the Muckenhoupt condition in the multidimensional case,” Visn. Odes. Nats. Univ., 12 , Issue 7, 80–84 (2007).MathSciNetGoogle Scholar
  5. 5.
    G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University, Cambridge (1934).Google Scholar
  6. 6.
    B. Bojarski, C. Sbordone, and I. Wik, “The Muckenhoupt class \( {A_1}\left( \mathbb{R} \right) \),” Stud. Math., 101 (2), 155–163 (1992).zbMATHMathSciNetGoogle Scholar
  7. 7.
    I. Klemes, “A mean oscillation inequality,” Proc. Amer. Math. Soc., 93, No. 3, 497–500 (1985).zbMATHMathSciNetGoogle Scholar
  8. 8.
    E. Yu. Leonchik and N. A. Malaksiano, “Exact indices of summability for functions from the classes A ,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 2, 17–26 (2007).Google Scholar
  9. 9.
    A. P. Calderón and A. Zygmund, “On the existence of certain singular integrals,” Acta Math., 88, 85–139 (1952).zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    A. A. Korenovskyy, A. K. Lerner, and A. M. Stokolos, “On a multidimensional form of F. Riesz’s “rising sun” lemma,” Proc. Amer. Math. Soc., 133, No. 5, 1437–1440 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    J. B. Garnett, Bounded Analytic Functions [Russian translation], Mir, Moscow (1984).zbMATHGoogle Scholar
  12. 12.
    E. M. Stein, Singular Integrals and Differential Properties of Functions, Princeton University, Princeton (1970).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • E. Yu. Leonchik
    • 1
  1. 1.Mechnikov Odessa National UniversityOdessaUkraine

Personalised recommendations